Mathematical imaging is not only a multidisciplinary research area but also a major cross-discipline subject within mathematical sciences as image analysis techniques involve differential geometry, optimization, nonlinear partial differential equations (PDEs), mathematical analysis, computational algorithms and numerical analysis. Segmentation refers to the essential problem in imaging and vision of automatically detecting objects in an image.

In this talk I first review some various models and techniques in the variational framework that are used for segmentation of images, with the purpose of discussing the state of arts rather than giving a comprehensive survey. Then I introduce the practically demanding task of detecting local features in a large image and our recent segmentation methods using energy minimization and PDEs. To ensure uniqueness and fast solution, we reformulate one non-convex functional as a convex one and further consider how to adapt the additive operator splitting method for subsequent solution. Finally I show our preliminary work to attempt segmentation of blurred images in the framework of joint deblurring and segmentation.